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Unformatted text preview: 1 Lecture 20 Finally, we remark that, as in the case of a real r , for complex r; D [ e rx ] = re rx : Indeed, let r = a + bi . Then D [ e rx ] = d dx & e ( a + bi ) x = d dx & e ax e ibx = d dx [ e ax (cos bx + i sin bx )] = ae ax (cos bx + i sin bx ) + e ax ( & b sin bx + bi cos bx ) = ae ax (cos bx + i sin bx ) + bie ax (cos bx + i sin bx ) = ( a + bi ) e ax (cos bx + i sin bx ) = re rx : Fundamental set of solutions By using that D [ e rx ] = re rx , as for real r in cases 1 and 2, we conclude that each complex r = a + bi of multiplicity m contributes to general solution & C + C 1 x + ::: + C m & 1 x m & 1 e r x : If r is a complex root of the characteristic equation, then r is also a complex root of the characteristic equation of the same multiplicity (this is because coe cients of the characteristic equation are real). So, r contributes to general solution & C + C 1 x + ::: + C m & 1 x m & 1 e r x : Notice that by Euler&s formula, e y 1 = & C + C 1 x + ::: + C m & 1...
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This note was uploaded on 01/26/2012 for the course MATH 285 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Differential Equations, Equations

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